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Find the velocity vector v(t) given the position vector r(t) = (((t)/(2))i + ((t)/(2))j) m.

April 13, 2026Physics

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Step-by-step solution

3 steps

Given the position vector:

\vec{r}(t) = \left(\cos\left(\frac{t}{2}\right)\hat{i} + \sin\left(\frac{t}{2}\right)\hat{j}\right) \, \text{m}

To differentiate this vector, we differentiate each component with respect to $t$ . This will give us the velocity vector $\vec{v}(t)$ .

Differentiate the $\hat{i}$ component.
Using the chain rule, $\frac{d}{dt}(\cos(kt)) = -k\sin(kt)$ . Here, $k = \frac{1}{2}$ .
$\frac{d}{dt}\left(\cos\left(\frac{t}{2}\right)\right) = -\frac{1}{2}\sin\left(\frac{t}{2}\right)$
Differentiate the $\hat{j}$ component.
Using the chain rule, $\frac{d}{dt}(\sin(kt)) = k\cos(kt)$ . Here, $k = \frac{1}{2}$ .
$\frac{d}{dt}\left(\sin\left(\frac{t}{2}\right)\right) = \frac{1}{2}\cos\left(\frac{t}{2}\right)$
Combine the differentiated components to form the velocity vector.
$\vec{v}(t) = \frac{d\vec{r}}{dt} = \left(-\frac{1}{2}\sin\left(\frac{t}{2}\right)\hat{i} + \frac{1}{2}\cos\left(\frac{t}{2}\right)\hat{j}\right) \, \text{m/s}$ $\boxed{\vec{v}(t) = \frac{1}{2}\left(-\sin\left(\frac{t}{2}\right)\hat{i} + \cos\left(\frac{t}{2}\right)\hat{j}\right) \, \text{m/s}}$